Distributions for two-point statistics of scattered signals with variable scattering strength

This article describes the distribution of complex products of the amplitudes of randomly scattered signals observed at two points. The signals at the two points are assumed to be fully saturated (zero-mean real and imaginary parts) and partially correlated, and to have a variance that is randomly modulated by the environment (e.g. turbulence, vegetation, or terrain variations). These assumptions are shown to lead to a new distribution for the real and imaginary parts of the complex product, called the compound variance gamma (CVΓ), which involves a Gauss hypergeometric function. The CVΓ distribution generalizes the compound gamma (CΓ), which applies to one-point statistics with randomly modulated scattering, and the variance gamma (VΓ), which applies to two-point statistics without random modulation. The CVΓ is shown to provide an excellent fit to two-point statistics derived from realistic simulations of sound propagation in the atmosphere including turbulent scattering, refraction by wind and temperature gradients, and ground reflections.